Chapter 7.4, Problem 92PPE

To determine

To find: The reason that the value of $0^0$ is undefined by using the division property of exponents.

Explanation of Solution

Given information: The division property of exponent for the expression $0^0$ .

The expression $z^z$ approaches $1$ as the positive value $z$ gets arbitrary close to zero does not prove that $0^0=1$ . The variable $z$ having a value close to zero is different than it having a value exactly zero. It turns the value of $0^0$ undefined.

To prove it is undefined, consider the value of $0^0$ is:

  $0^0=0^{1-1}$

Apply the property of addition of exponent of same base $\left(a^{m-n}=a^m\cdot a^{-n}\right)$ on the right hand side of the above expression.

  $0^0=0^1\cdot 0^{-1}$

Apply the inverse property of exponent $\left(a^{-n}=\frac{1}{a^n}\right)$ on the right hand side of the above expression.

  $\begin{array}{c}{0}^0=0^1\cdot \frac{1}{0^1}\\ =0\times \text{not}\text{defined}\\ =\text{undefined}\end{array}$

Therefore, from the above expression it is proved that $0^0$ equals to undefined.